Editing Picard–Lindelöf theorem

{{Short description|Existence and uniqueness of solutions to initial value problems}}
{{Differential equations}}

In [[mathematics]], specifically the study of [[differential equation]]s, the '''Picard–Lindelöf theorem''' gives a set of conditions under which an [[initial value problem]] has a [[Uniqueness (mathematics)|unique]] solution. It is also known as '''Picard's existence theorem''', the '''Cauchy–Lipschitz theorem''', or the '''existence and uniqueness theorem'''.

The theorem is named after [[Émile Picard]], [[Ernst Lindelöf]], [[Rudolf Lipschitz]] and [[Augustin-Louis Cauchy]].

== Theorem ==
Let  <math>D \subseteq \R \times \R^n</math> be a closed rectangle with <math>(t_0, y_0) \in \operatorname{int} D</math>, the interior of <math>D</math>. Let <math>f: D \to \R^n</math> be a function that is [[Continuous function|continuous]] in <math>t</math> and [[Lipschitz_continuity|Lipschitz continuous]] in <math>y</math> (with Lipschitz constant independent from <math>t</math>). Then there exists some <math>\varepsilon > 0</math> such that the initial value problem
<math display="block">y'(t)=f(t,y(t)),\qquad y(t_0)=y_0</math>
has a unique solution <math>y(t)</math> on the interval <math>[t_0-\varepsilon, t_0+\varepsilon]</math>.<ref>{{harvtxt|Coddington|Levinson|1955}}, Theorem I.3.1</ref><ref>{{Cite book |last1=Murray |first1=Francis |title=Existence Theorems for Ordinary Differential Equations |last2=Miller |first2=Kenneth |isbn= |pages=50}}</ref>

== Proof sketch ==
A standard proof relies on transforming the differential equation into an integral equation, then applying the [[Banach fixed-point theorem]] to prove the existence and uniqueness of solutions.

Integrating both sides of the differential equation <math display="inline">y'(t)=f(t,y(t))</math> shows that any solution to the differential equation must also satisfy the [[integral equation]]

:<math>y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds.</math>

Given the hypotheses that <math>f</math> is continuous in <math>t</math> and Lipschitz continuous in <math>y</math>, this integral operator is a [[Contraction (operator theory)|contraction]]{{why|date=February 2025}} and so the Banach fixed-point theorem proves that a solution can be obtained by [[fixed-point iteration]] of successive approximations. In this context, this fixed-point iteration method is known as [[Picard iteration]].

Set
:<math>\varphi_0(t)=y_0</math>
and
:<math>\varphi_{k+1}(t)=y_0+\int_{t_0}^t f(s,\varphi_k(s))\,ds.</math>

It follows from the Banach fixed-point theorem that the sequence of "Picard iterates" <math display="inline">\varphi_k</math> is [[Limit of a sequence|convergent]] and that its limit is a solution to the original initial value problem:

:<math>\lim_{k\to \infty} \varphi_k(t) = y(t)</math>.

Since the Banach fixed-point theorem states that the fixed-point is unique, the solution found through this iteration is the unique solution to the differential equation given an initial value.

== Example of Picard iteration ==
[[File:Picard iteration example tan x.svg|thumb|Four Picard iteration steps and their limit]]
Let <math>y(t)=\tan(t),</math> the solution to the equation <math>y'(t)=1+y(t)^2</math> with initial condition <math>y(t_0)=y_0=0,t_0=0.</math> Starting with <math>\varphi_0(t)=0,</math> we iterate 

:<math>\varphi_{k+1}(t)=\int_0^t (1+(\varphi_k(s))^2)\,ds</math> 

so that <math> \varphi_n(t) \to y(t)</math>:

:<math>\varphi_1(t)=\int_0^t (1+0^2)\,ds = t</math>

:<math>\varphi_2(t)=\int_0^t (1+s^2)\,ds = t + \frac{t^3}{3}</math>

:<math>\varphi_3(t)=\int_0^t \left(1+\left(s + \frac{s^3}{3}\right)^2\right)\,ds = t + \frac{t^3}{3} + \frac{2t^5}{15} + \frac{t^7}{63}</math>

and so on. Evidently, the functions are computing the [[Taylor series]] expansion of our known solution <math>y=\tan(t).</math> Since <math>\tan</math> has poles at <math>\pm\tfrac{\pi}{2},</math> it is not Lipschitz continuous in the neighborhood of those points, and the iteration converges toward a local solution for <math>|t|<\tfrac{\pi}{ 2}</math> only that is not valid over all of <math>\R</math>.

== Example of non-uniqueness ==
To understand uniqueness of solutions, contrast the following two examples of first order ordinary differential equations for {{math|''y''(''t'')}}.<ref>{{cite book |first=V. I. |last=Arnold |authorlink=Vladimir Arnold |title=Ordinary Differential Equations |publisher=The MIT Press |year=1978 |isbn=0-262-51018-9 }}</ref> Both differential equations will possess a single stationary point {{math|''y'' {{=}} 0.}} 

First, the homogeneous linear equation {{math|{{sfrac|''dy''|''dt''}} {{=}} ''ay''}} (<math>a<0</math>), a stationary solution is {{math|''y''(''t'') {{=}} 0}}, which is obtained for the initial condition {{math|''y''(0) {{=}} 0}}. Beginning with any other initial condition {{math|''y''(0) {{=}} ''y''<sub>0</sub> ≠ 0}}, the solution <math>y(t) = y_0 e^{at}</math> tends toward the stationary point {{math|''y'' {{=}} 0}}, but it only approaches it in the limit of infinite time, so the uniqueness of solutions over all finite times is guaranteed. 

By contrast for an equation in which the stationary point can be reached after a ''finite'' time, uniqueness of solutions does not hold. Consider the homogeneous nonlinear equation {{math|{{sfrac|''dy''|''dt''}} {{=}} ''ay''<sup>&thinsp;{{sfrac|2|3}}</sup>}}, which has at least these two solutions corresponding to the initial condition {{math|''y''(0) {{=}} 0}}: {{math|''y''(''t'') {{=}} 0}} and 

:<math>y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\ \ \ \ \ 0 & t \ge 0, \end{cases}</math>

so the previous state of the system is not uniquely determined by its state at or after ''t'' = 0. The uniqueness theorem does not apply because the derivative of the function {{math|&thinsp;''f''&thinsp;(''y'') {{=}} ''y''<sup>&thinsp;{{sfrac|2|3}}</sup>}} is not bounded in the neighborhood of {{math|''y'' {{=}} 0}} and therefore it is not Lipschitz continuous, violating the hypothesis of the theorem.

== Detailed proof ==

Let <math>L</math> be the Lipschitz constant of <math>(t, y) \mapsto f(t,y)</math> with respect to <math>y.</math>
The function <math>f</math> is continuous as a function of <math>(t,y)</math>.
In particular, since <math>t \mapsto f(t,y)</math> is a continuous function of <math>t</math>, we have that for any point <math>(t_0, y_0)</math> and <math>\epsilon>0</math>  there exist <math>\delta>0</math>
such that <math>|f(t,y_0)-f(t_0,y_0)| <\epsilon / 2</math> when <math>|t - t_0| < \delta</math>. 
We have
<math display="block"> |f(t,y)-f(t_0,y_0)|\leq |f(t,y)-f(t,y_0)|+|f(t,y_0)-f(t_0,y_0)|<\epsilon, </math>
provided <math>|t-t_0|<\delta</math> and <math>|y-y_0|<\epsilon /2L</math>, which shows that <math>f</math> is continuous at <math>(t_0,y_0)</math>.

Let <math>a := 1/2L</math> and take any <math>b > 0</math> such that 
<math display="block"> C_{a,b} = I_a(t_0) \times B_b(y_0) </math>
is a subset of <math>D,</math> where
<math display="block">\begin{align}
I_a(t_0) &= [t_0-a,t_0+a] \\
B_b(y_0) &= [y_0-b,y_0+b].
\end{align}</math>
Such a set exists because <math>(t_0, y_0)</math> is in the interior of <math>D,</math> by assumption. 
<!--<math>C_{a,b}</math>  is a compact rectangular set where &thinsp;{{math|''f''}}&thinsp; is defined. -->

Let

:<math>M = \sup_{(t,y) \in C_{a,b}}\|f(t,y)\|,</math>

which is the [[supremum]] of (the [[absolute value]]s of) the slopes of the function.
The function <math>f</math> attains a maximum on <math>C_{a,b}</math> because <math>f</math> is continuous and <math>C_{a,b}</math> is compact. 
For a later step in the proof, we need that <math>a < b / M,</math> so if <math>a \geq b / M,</math> then change <math>a</math> to <math>a :=\tfrac{1}{2}\min\{1 / L,\ b / M\},</math> and update <math>I_{a}(t_0),</math> <math>B_{b}(y_0),</math> <math>C_{a,b},</math> and <math>M</math> accordingly (this update will be needed at most once since <math>M</math> cannot increase as a result of restricting <math>C_{a,b}</math>).

Consider <math>\mathcal{C}(I_{a}(t_0),B_b(y_0))</math>, the [[function space]] of continuous functions <math>I_{a}(t_0)\to B_b(y_0).</math>
We will proceed by applying the [[Banach fixed-point theorem]] using the [[metric (mathematics)|metric]] on <math>\mathcal{C}(I_{a}(t_0),B_b(y_0))</math> induced by the [[uniform norm]]. Namely, for each continuous function <math>\varphi :  I_{a}(t_0) \to B_b(y_0),</math> the norm of <math>\varphi</math> is
<math display="block">\| \varphi \|_\infty = \sup_{t \in I_a} \| \varphi(t)\|.</math>
<!--\varphi should be a function of t and x, but it is only written as a function of t!-->
The ''Picard operator'' <math display="block">\Gamma:\mathcal{C}\big(I_{a}(t_0),B_b(y_0)\big) \to \mathcal{C}\big(I_{a}(t_0),B_b(y_0)\big)</math> is defined for each <math>\varphi \in \mathcal{C}(I_{a}(t_0),B_b(y_0))</math> by <math>\Gamma \varphi \in \mathcal{C}(I_{a}(t_0),B_b(y_0))</math> given by
<math display="block">\Gamma \varphi(t) = y_0 + \int_{t_0}^{t} f(s,\varphi(s)) \, ds \quad \forall t \in I_a(t_0).</math>

To apply the Banach fixed-point theorem, we must show that <math>\Gamma</math> maps a complete non-empty [[metric space]] ''X'' into itself and also is a [[contraction mapping]].

We first show that <math>\Gamma</math> takes <math>B_b(y_0)</math> into itself in the space of continuous functions with the uniform norm. 
<!-- The statement "<math>\Gamma</math> takes <math>\overline{B_b(y_0)}</math> into itself" is not technically correct, since \Gamma acts on functions *and* those functions should have \mathcal{C}(I_{a}(t_0),B_b(y_0)) as their domain.-->
Here, <math>B_b(y_0)</math> is a closed ball in the space of continuous (and [[bounded function|bounded]]) functions "centered" at the constant function <math>y_0</math>. 
Hence we need to show that 
<math display="block>\| \varphi -y_0 \|_\infty \le b</math>
implies
<math display="block>\left\| \Gamma\varphi(t)-y_0 \right\| = \left\|\int_{t_0}^t f(s,\varphi(s))\, ds \right\| \leq \int_{t_0}^{t'} \left\|f(s,\varphi(s))\right\| ds \leq \int_{t_0}^{t'} M\, ds = M \left|t'-t_0 \right| \leq M a \leq b</math>

where <math>t'</math> is some number in <math>[t_0-a, t_0 +a]</math> where the maximum is achieved. 
The last inequality in the chain is true since <math>a < b / M.</math>
<!--To-do: This preceding paragraph mixes a statement of what we "want to show" with the actual thing we are showing.--> 

Now let us prove that <math>\Gamma</math> is a contraction mapping as required to apply the [[Banach fixed-point theorem]].
In particular, we want to show that there exists <math>0 \leq q < 1,</math> such that
<math display="block"> \left \| \Gamma \varphi_1 - \Gamma \varphi_2 \right\|_\infty \le q  \left\| \varphi_1 - \varphi_2 \right\|_\infty</math>
for all <math>\varphi_1,\varphi_2\in\mathcal{C}(I_{a}(t_0),B_b(y_0)).</math>

Let <math>q = aL</math> and take any <math>\varphi_1,\varphi_2\in\mathcal{C}(I_{a}(t_0),B_b(y_0)).</math>
Take <math>t</math> such that

:<math>\| \Gamma \varphi_1 - \Gamma \varphi_2 \|_\infty =  \left\| \left(\Gamma\varphi_1 - \Gamma\varphi_2 \right)(t) \right\|.</math>
<!-- To-do: Explain why such a t exists -->

Then, using the definition of <math>\Gamma</math>,

:<math>\begin{align}
\left\|\left(\Gamma\varphi_1 - \Gamma\varphi_2 \right)(t) \right\| &= \left\|\int_{t_0}^t \left( f(s,\varphi_1(s))-f(s,\varphi_2(s)) \right)ds \right\|\\
&\leq \int_{t_0}^t \left\|f \left(s,\varphi_1(s)\right)-f\left(s,\varphi_2(s) \right) \right\| ds \\
&\leq L \int_{t_0}^t \left\|\varphi_1(s)-\varphi_2(s) \right\|ds  && \text{since } f \text{ is Lipschitz-continuous} \\
&\leq L \int_{t_0}^t \left\|\varphi_1-\varphi_2 \right\|_\infty \,ds \\
&\leq La \left\|\varphi_1-\varphi_2 \right\|_\infty,
\end{align}</math>
where <math>t - t_0 \leq a,</math> because the domains of <math>\phi_1,\phi_2</math> are both <math>I_a(t_0) \times B_b(y_0).</math>
By definition, <math>q = aL,</math> and <math>a < 1 / L,</math> so <math>q < 1.</math>
Therefore, <math>\Gamma</math> is a contraction.
<!--To-do: This proof should construct a value of q \in [0, 1) such that |\Gamma \phi_1 - \Gamma \phi_2| \leq q |\phi_1 - \phi_2|_\infty for all \phi_1, \phi_2. -->

We have established that the Picard's operator is a contraction on the [[Banach space]]s with the metric induced by the uniform norm. This allows us to apply  the Banach fixed-point theorem to conclude that the operator has a unique fixed point. In particular, there is a unique function
<math display="bl">\varphi\in \mathcal{C}(I_a (t_0), B_b(y_0))</math>
such that 
<math display="block">\Gamma \varphi = \varphi.</math> 
Thus, <math>\varphi</math> is the unique solution of the initial value problem, valid on the interval <math>I_a.</math>

==Optimization of the solution's interval==
We wish to remove the dependence of the interval ''I<sub>a</sub>'' on ''L''. To this end, there is a [[corollary]] of the Banach fixed-point theorem: if an operator ''T''<sup>''n''</sup> is a contraction for some ''n'' in '''N''', then ''T'' has a unique fixed point. Before applying this theorem to the Picard operator, recall the following:

{{math theorem | name = Lemma | math_statement =
<math>\left\| \Gamma^m \varphi_1(t) - \Gamma^m\varphi_2(t) \right\| \leq \frac{L^m|t-t_0|^m}{m!}\left\|\varphi_1-\varphi_2\right\|</math> for all <math>t \in [t_0 - \alpha, t_0 + \alpha]</math>
}}

''Proof.'' [[Mathematical induction|Induction]] on ''m''. For the base of the induction ({{math|1=''m'' = 1}}) we have already seen this, so suppose the inequality holds for {{math|''m'' − 1}}, then we have: 
<math display="block">\begin{align}
\left \| \Gamma^m \varphi_1(t) - \Gamma^m\varphi_2(t) \right \| &= \left \|\Gamma\Gamma^{m-1} \varphi_1(t) - \Gamma\Gamma^{m-1}\varphi_2(t) \right \| \\
&\leq \left| \int_{t_0}^t \left \| f \left (s,\Gamma^{m-1}\varphi_1(s) \right )-f \left (s,\Gamma^{m-1}\varphi_2(s) \right )\right \| ds \right| \\
&\leq L \left| \int_{t_0}^t  \left \|\Gamma^{m-1}\varphi_1(s)-\Gamma^{m-1}\varphi_2(s)\right \| ds\right| \\
&\leq L \left| \int_{t_0}^t  \frac{L^{m-1}|s-t_0|^{m-1}}{(m-1)!}  \left \| \varphi_1-\varphi_2\right \| ds\right| \\
&\leq \frac{L^m |t-t_0|^m }{m!} \left \|\varphi_1 - \varphi_2 \right \|.
\end{align}</math>

By taking a supremum over <math> t \in [t_0 - \alpha, t_0 + \alpha] </math> we see that <math>\left \| \Gamma^m \varphi_1 - \Gamma^m\varphi_2 \right \| \leq \frac{L^m\alpha^m}{m!}\left \|\varphi_1-\varphi_2\right \|</math>.

This inequality assures that for some large ''m'', 
<math display="block">\frac{L^m\alpha^m}{m!}<1,</math>
and hence Γ<sup>''m''</sup> will be a contraction. So by the previous corollary Γ will have a unique fixed point. Finally, we have been able to optimize the interval of the solution by taking {{math|1=''α'' = min{''a'', {{sfrac|''b''|''M''}}}<nowiki/>}}.

In the end, this result shows the interval of definition of the solution does not depend on the Lipschitz constant of the field, but only on the interval of definition of the field and its maximum absolute value.

== Other existence theorems ==
The Picard–Lindelöf theorem shows that the solution exists and that it is unique. The [[Peano existence theorem]] shows only existence, not uniqueness, but it assumes only that {{math|&thinsp;''f''&thinsp;}} is continuous in {{mvar|y}}, instead of [[Lipschitz continuous]]. For example, the right-hand side of the equation {{math|{{sfrac|''dy''|''dt''}} {{=}} ''y''<sup>&thinsp;{{sfrac|1|3}}</sup>}} with initial condition {{nowrap|1=''y''(0) = 0}} is continuous but not Lipschitz continuous. Indeed, rather than being unique, this equation has at least three solutions:<ref>{{harvtxt|Coddington|Levinson|1955}}, p. 7</ref>

:<math>y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}</math>.

Even more general is [[Carathéodory's existence theorem]], which proves existence (in a more general sense) under weaker conditions on {{math|&thinsp;''f''&thinsp;}}. Although these conditions are only sufficient, there also exist necessary and sufficient conditions for the solution of an initial value problem to be unique, such as [[Hiroshi Okamura|Okamura]]'s theorem.<ref>{{cite book |first1=Ravi P. |last1=Agarwal |first2=V. |last2=Lakshmikantham |title=Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations |url=https://books.google.com/books?id=q4OkW4H8BCUC&pg=PA159 |year=1993 |publisher=World Scientific |isbn=981-02-1357-3 |page=159 }}</ref>

== Global existence of solution ==
The Picard–Lindelöf theorem ensures that solutions to initial value problems exist uniquely within a local interval <math>[t_0-\varepsilon, t_0+\varepsilon]</math>, possibly dependent on each solution. The behavior of solutions beyond this local interval can vary depending on the properties of {{math|&thinsp;''f''&thinsp;}} and the domain over which {{math|&thinsp;''f''&thinsp;}} is defined. For instance, if {{math|&thinsp;''f''&thinsp;}} is globally Lipschitz, then the local interval of existence of each solution can be extended to the entire real line and all the solutions are defined over the entire '''R'''. 

If {{math|&thinsp;''f''&thinsp;}} is only locally Lipschitz, some solutions may not be defined for certain values of ''t'', even if {{math|&thinsp;''f''&thinsp;}} is smooth. For instance, the differential equation {{math|{{sfrac|''dy''|''dt''}} {{=}} ''y''<sup>&thinsp;2</sup>}} with initial condition {{nowrap|1=''y''(0) = 1}} has the solution ''y''(''t'') = 1/(1-''t''), which is not defined at ''t'' = 1. Nevertheless, if {{math|&thinsp;''f''&thinsp;}} is a differentiable function defined on a compact submanifold of '''R'''<sup>n</sup> such that the prescribed derivative is tangent to the given submanifold, then the initial value problem has a unique solution for all time. More generally, in [[differential geometry]]: if {{math|&thinsp;''f''&thinsp;}} is a differentiable [[vector field]] defined over a domain which is a [[Compact manifold|compact smooth manifold]], then all its trajectories ([[Integral curve|integral curves]]) exist for all time.<ref name=":0">{{Cite book |last=Perko |first=Lawrence Marion |title=Differential equations and dynamical systems |publisher=Springer |year=2001 |isbn=978-1-4613-0003-8 |edition=3rd |series=Texts in applied mathematics |location=New York |pages=189}}</ref><ref>{{Citation |last=Lee |first=John M. |title=Smooth Manifolds |date=2003 |work=Introduction to Smooth Manifolds |series=Graduate Texts in Mathematics |volume=218 |pages= 1–29|url=http://dx.doi.org/10.1007/978-0-387-21752-9_1 |access-date= |place=New York, NY |publisher=Springer New York |doi=10.1007/978-0-387-21752-9_1 |isbn=978-0-387-95448-6|url-access=subscription }}</ref>

== See also ==
{{Portal|Mathematics}}
* [[Cauchy–Kovalevskaya theorem]]
* [[Vector field#Complete vector fields|Complete vector fields]]
* [[Frobenius theorem (differential topology)]]
* [[Integrability conditions for differential systems]]
* [[Newton's method]]
* [[Euler method]]
* [[Trapezoidal rule]]

== Notes ==
{{Reflist}}

== References ==
{{refbegin}}
* {{Cite book | last1=Coddington | first1=Earl A. |authorlink=Earl A. Coddington | last2=Levinson | first2=Norman |authorlink2=Norman Levinson | title=Theory of Ordinary Differential Equations | publisher=[[McGraw-Hill]] | year=1955 |isbn=9780070992566}}
* {{cite journal |first=E. |last=Lindelöf |title=Sur l'application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre |journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences |volume=118 |year=1894 |pages=454–7 |url=http://gallica.bnf.fr/ark:/12148/bpt6k3074r/f454.table }} (In that article Lindelöf discusses a generalization of an earlier approach by Picard.)
* {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island]]|series=[[Graduate Studies in Mathematics]] |issn=1065-7339 |eissn=2376-9203 | year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/|section=2.2. The basic existence and uniqueness result|section-url=https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf#page=47 |page=38 |zbl=1263.34002 |lang=en}}
{{refend}}

== External links ==
{{refbegin}}
*{{cite encyclopedia |url=https://www.encyclopediaofmath.org/index.php/Cauchy-Lipschitz_theorem |title=Cauchy-Lipschitz theorem |encyclopedia=[[Encyclopedia of Mathematics]]}}
* [https://web.archive.org/web/20100616152308/http://www.krellinst.org/UCES/archive/classes/CNA/dir2.6/uces2.6.html Fixed Points and the Picard Algorithm], recovered from http://www.krellinst.org/UCES/archive/classes/CNA/dir2.6/uces2.6.html.
* {{cite web |url=http://www.math.byu.edu/~grant/courses/m634/f99/lec4.pdf |first=Christopher |last=Grant |title=Lecture 4: Picard-Lindelöf Theorem |date=1999 |work=Math 634: Theory of Ordinary Differential Equations |publisher=Department of Mathematics, Brigham Young University }}
{{refend}}

{{Differential equations topics}}

{{DEFAULTSORT:Picard-Lindelof theorem}}
[[Category:Augustin-Louis Cauchy]]
[[Category:Lipschitz maps]]
[[Category:Ordinary differential equations]]
[[Category:Theorems in mathematical analysis]]
[[Category:Uniqueness theorems]]